Summary: We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack [Classical Quantum Gravity 24, No. 4, 809–828 (2007; Zbl 1111.83002)] in the case of closed manifolds and establish existence and non-existence results for constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.